Differential abundance analysis is at the core of statistical analysis of microbiome data. The compositional nature of microbiome sequencing data makes false positive control challenging. Here, we show that the compositional effects can be addressed by a simple, yet highly flexible and scalable, approach. The proposed method, LinDA, only requires fitting linear regression models on the centered log-ratio transformed data, and correcting the bias due to compositional effects. We show that LinDA enjoys asymptotic FDR control and can be extended to mixed-effect models for correlated microbiome data. Using simulations and real examples, we demonstrate the effectiveness of LinDA.

Non-steroidal anti-inflammatory drugs (NSAIDs) are among the most frequently used classes of medications in the world, yet they induce an enteropathy that is associated with high morbidity and mortality. A major limitation to better understanding the pathophysiology and diagnosis of this enteropathy is the difficulty of obtaining information about the primary site of injury, namely the distal small intestine. We investigated the utility of using mRNA from exfoliated cells in stool as a means to surveil the distal small intestine in a murine model of NSAID enteropathy. Specifically, we performed RNA-Seq on exfoliated cells found in feces and compared these data to RNA-Seq from both the small intestinal mucosa and colonic mucosa of healthy control mice or those exhibiting NSAID-induced enteropathy. Global gene expression analysis, data intersection, pathway analysis, and computational approaches including linear discriminant analysis (LDA) and sparse canonical correlation analysis (CCA) were used to assess the inter-relatedness of tissue (invasive) and stool (noninvasive) datasets. These analyses revealed that the exfoliated cell transcriptome closely mirrored the transcriptome of the small intestinal mucosa. Thus, the exfoliome may serve as a non-invasive means of detecting and monitoring NSAID enteropathy (and possibly other gastrointestinal mucosal inflammatory diseases).

We propose a novel use of a broadcasting operation, which distributes univariate functions to all entries of the tensor covariate, to model the nonlinearity in tensor regression nonparametrically. A penalized estimation and the corresponding algorithm are proposed. Our theoretical investigation, which allows the dimensions of the tensor covariate to diverge, indicates that the proposed estimation yields a desirable convergence rate. We also provide a minimax lower bound, which characterizes the optimality of the proposed estimator for a wide range of scenarios. Numerical experiments are conducted to confirm the theoretical findings, and they show that the proposed model has advantages over its existing linear counterparts.

Motivated by the study of gene and environment interactions, we consider a multivariate response varying-coefficient model with a large number of covariates. The need of nonparametrically estimating a large number of coefficient functions given relatively limited data poses a big challenge for fitting such a model. To overcome the challenge, we develop a method that incorporates three ideas: (i) reduce the number of unknown functions to be estimated by using (noncentered) principal components; (ii) approximate the unknown functions by polynomial splines; (iii) apply sparsity-inducing penalization to select relevant covariates. The three ideas are integrated into a penalized least-square framework. Our asymptotic theory shows that the proposed method can consistently identify relevant covariates and can estimate the corresponding coefficient functions with the same convergence rate as when only the relevant variables are included in the model. We also develop a novel computational algorithm to solve the penalized least-square problem by combining proximal algorithms and optimization over Stiefel manifolds. Our method is illustrated using data from Framingham Heart Study.

Multivariate functional data arise in a wide range of applications. One fundamental task is to understand the causal relationships among these functional objects of interest. In this paper, we develop a novel Bayesian network (BN) model for multivariate functional data where conditional independencies and causal structure are encoded by a directed acyclic graph. Specifically, we allow the functional objects to deviate from Gaussian processes, which is the key to unique causal structure identification even when the functions are measured with noises. A fully Bayesian framework is designed to infer the functional BN model with natural uncertainty quantification through posterior summaries. Simulation studies and real data examples demonstrate the practical utility of the proposed model.

Tensor regression is an important and useful tool for analyzing multidimensional array data. To deal with high dimensionality, CANDECOMP/PARAFAC (CP) low-rank constraints are often imposed on the coefficient tensor parameter in the (penalized) loss functions. However, besides the well-known non-identifiability issue of the CP parameters, we demonstrate that the corresponding optimization may not have any attainable solutions, and thus the estimation of the coefficient tensor is not well-defined when this happens. This is closely related to a phenomenon, called CP degeneracy, in low-rank tensor approximation problems. In this article, we show some useful results of CP degeneracy in the context of tensor regression problems. To overcome the theoretical and numerical issues associated with the degeneracy, we provide a general penalized strategy as a solution to the degeneracy. The related results also explain why some of the existing methods are more stable than the others. The asymptotic properties of the resulting estimation are also studied. Numerical experiments are conducted to illustrate our findings.